Can a single neural network handle recognizing two types of objects, or should it be split into two smaller networks?
Well, I do not know what type of features you are giving to your neural network. However, in general, I would go with a single neural network. It seems that you have no limitation in resources for training your network and the only problem is resources while you apply your network. 

The thing is that probably the two problems have things in common (e.g. both types of plates are rectangular). This means that if you use two networks, each has to solve the same sub-problem (the common part) again. If you use only one network the common part of the problem takes fewer cells/weights to be solved and the remaining weights/cells can be employed for better recognition.

In the end, if I was in your place I would try both of them. I think that is the only way to be really sure what is the best solution. When speaking theoretically it is possible that we do not include some factors.